Problem: $ C = \left[\begin{array}{rr}2 & 5 \\ 3 & 1\end{array}\right]$ $ B = \left[\begin{array}{rrr}3 & 0 & -2 \\ 5 & -1 & 3\end{array}\right]$ What is $ C B$ ?
Solution: Because $ C$ has dimensions $(2\times2)$ and $ B$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ C B = \left[\begin{array}{rr}{2} & {5} \\ {3} & {1}\end{array}\right] \left[\begin{array}{rrr}{3} & \color{#DF0030}{0} & \color{#9D38BD}{-2} \\ {5} & \color{#DF0030}{-1} & \color{#9D38BD}{3}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ B$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ B$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ B$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{2}\cdot{3}+{5}\cdot{5} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ B$ and add the products together. $ \left[\begin{array}{rrr}{2}\cdot{3}+{5}\cdot{5} & ? & ? \\ {3}\cdot{3}+{1}\cdot{5} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ C$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ B$ and add the products together. $ \left[\begin{array}{rrr}{2}\cdot{3}+{5}\cdot{5} & {2}\cdot\color{#DF0030}{0}+{5}\cdot\color{#DF0030}{-1} & ? \\ {3}\cdot{3}+{1}\cdot{5} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{2}\cdot{3}+{5}\cdot{5} & {2}\cdot\color{#DF0030}{0}+{5}\cdot\color{#DF0030}{-1} & {2}\cdot\color{#9D38BD}{-2}+{5}\cdot\color{#9D38BD}{3} \\ {3}\cdot{3}+{1}\cdot{5} & {3}\cdot\color{#DF0030}{0}+{1}\cdot\color{#DF0030}{-1} & {3}\cdot\color{#9D38BD}{-2}+{1}\cdot\color{#9D38BD}{3}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}31 & -5 & 11 \\ 14 & -1 & -3\end{array}\right] $